10 problems

Let $(V,c)$ and the matrix $(A_{ij})$ be as above: $V=\mathbb C x_1\oplus\cdots\oplus\mathbb C x_{\theta}$, $c(x_i\otimes x_j)=q_{ij}x_j\otimes x_i$, $q_{ii}=-1$,…

Let $(V,c)$ be a braided vector space with $V=\mathbb C x_1\oplus\cdots\oplus\mathbb C x_{\theta}$ and … Assume that $q_{ii}=-1$ for every $i$ and $q_{ij}\in\{\pm1\}$ for all $i,j$…

Finite-GK-dimension conjecture. Nichols algebras of diagonal type with finite Gelfand–Kirillov dimension have finite root system.

Angiono–Azevedo–Heckenberger conjecture. If $\mathscr{B}_{\mathfrak{q}}$ has finite Gelfand–Kirillov dimension, then its root system is finite.

Let $V$ be a finite-dimensional braided vector space of diagonal type, and let $\mathscr{B}(V)$ denote its Nichols algebra. An arithmetic root system conjecture. If … then $V$ has…

Let $\mathfrak{q}$ be a braiding matrix of diagonal type, and let ${\mathcal B}_{\mathfrak{q}}$ be the corresponding Nichols algebra. A matrix $\mathfrak{q}$ is called arithmetic w…

Let $V$ be a finite-dimensional braided vector space of diagonal type, let $\mathcal{B}(V)$ be its Nichols algebra, and let the corresponding generalized root system be the root sy…

Let $(V,c)$ be a group-type BVS, and let $G=\langle g_i\rangle$ be the group generated by its associated operators. Assume that $c$ is unitary and has finite order. Finite-group or…

Finite-group or monomiality conjecture. Either $G$ is finite, or there is a basis with respect to which $c$ is monomial.

Virtually abelian image conjecture. For every $n$, the image $\rho^c(\mathcal{B}_n)$ is virtually abelian; moreover, if $c$ has finite order, then $\rho^c(\mathcal{B}_n)$ is finite…